# How do two Poisson Processes affect each other?

Widgets of Type A arrive with Poisson Process with arrival rate $\lambda_A$, and, for Type B, with arrival rate $\lambda_B$ (independent).

During t, there have been b arrivals of Type B. What are the expected arrivals of Type A+B in time frame t?

Does one simply take the given value b, and add to that the expected arrivals for process A?:

$b+t \times \lambda_A$

• Assuming that both processes are independent, you can do like that. otherwise, it is inconclusive unless some auxiliary conditions are assumed. – Sangchul Lee Oct 22 '12 at 6:56
• thanks but why do you not post this as an answer, but as a comment? – Wuschelbeutel Kartoffelhuhn Oct 22 '12 at 7:00
In general, $N_t=N^A_t+N^B_t$ implies $\mathbb E(N_t\mid N^B_t=b)=\mathbb E(N^A_t\mid N^B_t=b)+b$. If furthermore the processes $N^A$ and $N^B$ are independent, then $\mathbb E(N^A_t\mid N^B_t=b)=\mathbb E(N^A_t)=\lambda_At$.
Thus, in your setting, $\mathbb E(N_t\mid N^B_t=b)=b+\lambda_At$.